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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Operator ideal norms on $L^{p}$


Authors: L. Rodríguez-Piazza and M. C. Romero-Moreno
Journal: Trans. Amer. Math. Soc. 352 (2000), 379-395
MSC (1991): Primary 47D50, 46E30
Published electronically: July 20, 1999
MathSciNet review: 1473454
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Abstract: Let $p$ be a real number such that $p \in (1,+\infty )$ and its conjugate exponent $q\not =4,6,8\ldots $. We prove that for an operator $T$ defined on $L^{p}(\lambda )$ with values in a Banach space, the image of the unit ball determines whether $T$ belongs to any operator ideal and its operator ideal norm. We also show that this result fails to be true in the remaining cases of $p$. Finally we prove that when the result holds in finite dimension, the map which associates to the image of the unit ball the operator ideal norm is continuous with respect to the Hausdorff metric.


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Additional Information

L. Rodríguez-Piazza
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, Sevilla 41080, Spain
Email: piazza@cica.es

M. C. Romero-Moreno
Affiliation: Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad de Sevilla, Aptdo. 1160, Sevilla 41080, Spain
Email: mcromero@cica.es

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02196-0
PII: S 0002-9947(99)02196-0
Keywords: Operator ideals, ideal norm, $L^p$ spaces, Hausdorff metric, $p$-integral operators
Received by editor(s): May 30, 1997
Published electronically: July 20, 1999
Additional Notes: Research supported in part by DGICYT grant #PB93–0926
Article copyright: © Copyright 1999 American Mathematical Society



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